As the mean is E[X] =150 and [tex]\sigma [/tex] = 30, using the normal distribution to get the answer is just using Z. So, what we need is P(X<180).
[tex]Z= \frac{x-E[X]}{\sigma } = \frac{180-150}{30} = \frac{30}{30} = 1[/tex]
P(x<180) = P(z< 1) = [tex]\phi [/tex](1) = 0.8413.
Then, just multiplying the amount of books you have, which is 500, with the probability would give how many books are less than 180 pages, being:
[tex]0.8413 * 500 = 420.65[/tex]
